Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x + \cos x = \sqrt{2}.$  Enter all the solutions, separated by commas.
Squaring both sides, we get
\[\sin^2 x + 2 \sin x \cos x + \cos x^2 = 2.\]Then $2 \sin x \cos x = 1,$ so $\sin 2x = 1.$  Since $0 \le x < 2 \pi,$ $2x = \frac{\pi}{2}$ or $2x = \frac{5 \pi}{2},$ so $x = \frac{\pi}{4}$ or $x = \frac{5 \pi}{4}.$  We check that only $\boxed{\frac{\pi}{4}}$ works.